The performance and reliability of modern new energy vehicles are fundamentally dependent on robust and intelligent Battery Management Systems (BMS). A core, yet challenging, function of the BMS is the accurate, real-time estimation of the State-of-Charge (SOC) for the lithium-ion battery pack. SOC, defined as the ratio of remaining capacity to nominal capacity, is a critical parameter analogous to a fuel gauge. Its precise knowledge is indispensable for optimizing energy usage, preventing over-charge/discharge damage, predicting driving range, and ensuring the safe and efficient operation of the entire electric powertrain. However, the SOC of a lithium-ion battery is an internal state that cannot be measured directly and must be inferred from other measurable quantities such as terminal voltage, current, and temperature.
Traditional methods for lithium-ion battery SOC estimation each have significant limitations. The open-circuit voltage (OCV) method requires the battery to rest for an extended period to reach equilibrium, making it unsuitable for real-time estimation during operation. The ampere-hour (Ah) integration method is simple but suffers from accumulated errors due to current sensor drift, noise, and an unknown initial SOC. While a combination of these methods is often used in practice, it still lacks the dynamic robustness needed for complex operational profiles. Data-driven and machine learning approaches can model complex non-linearities but demand vast amounts of training data and substantial computational resources, raising concerns about real-time applicability and generalization.
Consequently, model-based estimation filters have become the predominant approach for online SOC estimation. Among these, algorithms based on Kalman Filter (KF) variants are highly regarded. However, standard KFs and Extended Kalman Filters (EKFs) require accurate a priori knowledge of the statistical properties of process and measurement noises, which are difficult to obtain precisely for a dynamic system like a lithium-ion battery under varying operating conditions. Inaccurate noise assumptions can lead to performance degradation or even filter divergence.
The H-infinity filter (HF) presents a powerful alternative. It operates on a min-max optimization criterion, aiming to minimize the worst-case estimation error without strict assumptions about noise statistics. This characteristic grants the HF algorithm superior robustness against modeling uncertainties and noise variations, making it particularly attractive for lithium-ion battery SOC estimation where model parameters change with SOC, temperature, and aging. Despite its robustness, a notable weakness of the standard HF is its relative insensitivity to sudden changes in the system’s state or model parameters. Its performance in tracking abrupt state transitions, which can occur during aggressive driving cycles or fault conditions, can be inadequate.
To address this specific limitation, we propose and develop a novel Strong Tracking H-infinity Filter (STHF). This hybrid algorithm synergistically combines the inherent robustness of the HF against noise uncertainties with the superior state-tracking capability of the Strong Tracking Filter (STF) concept. The core innovation lies in the introduction of a time-varying fading factor into the HF’s error covariance prediction step. This factor is calculated online based on the innovation sequence (the difference between measured and predicted outputs), effectively amplifying the filter’s gain when the actual system behavior diverges from the model prediction. This allows the STHF algorithm to rapidly recalibrate and maintain accurate tracking during dynamic transients, a scenario where the standard HF might lag. In this comprehensive article, we will detail the complete methodology: first, establishing an accurate electrical model for the lithium-ion battery; second, deriving the mathematical framework for both HF and the proposed STHF; and finally, validating the superior performance of the STHF algorithm against the standard HF through rigorous experimental testing under diverse drive cycles.
Electrical Modeling of the Lithium-Ion Battery
An accurate yet computationally efficient model is the cornerstone of any model-based estimation algorithm. For BMS applications, equivalent circuit models (ECMs) are favored due to their good balance between model fidelity and simplicity. We adopt the second-order RC ECM, which effectively captures the dominant dynamic behaviors of a lithium-ion battery.
The schematic of the second-order RC model is shown in Figure 1 (conceptually represented). It consists of an ideal voltage source \(U_{oc}(SOC)\), an ohmic internal resistor \(R_0\), and two parallel RC networks: \((R_1, C_1)\) and \((R_2, C_2)\). The open-circuit voltage \(U_{oc}\) is a nonlinear function of the SOC. The resistor \(R_0\) represents the instantaneous voltage drop. The first RC branch \((R_1, C_1)\) models the short-term electrochemical polarization dynamics, while the second branch \((R_2, C_2)\) models the long-term concentration polarization dynamics.
Applying Kirchhoff’s laws, the electrical behavior of this model is described by the following state-space equations:
$$
\begin{aligned}
\frac{dU_1}{dt} &= -\frac{1}{R_1 C_1}U_1 + \frac{1}{C_1}I \\
\frac{dU_2}{dt} &= -\frac{1}{R_2 C_2}U_2 + \frac{1}{C_2}I \\
U_t &= U_{oc}(SOC) – U_1 – U_2 – R_0 I
\end{aligned}
$$
where \(I\) is the load current (positive for discharge, negative for charge), \(U_t\) is the terminal voltage, and \(U_1\), \(U_2\) are the voltages across the polarization RC networks.
The SOC is governed by the ampere-hour integration principle:
$$
SOC(t) = SOC(t_0) – \frac{1}{C_n} \int_{t_0}^{t} \eta I(\tau) d\tau
$$
where \(C_n\) is the nominal capacity of the lithium-ion battery, and \(\eta\) is the Coulombic efficiency (often assumed to be 1 for discharge and a constant slightly less than 1 for charge).
To implement a discrete-time filter, we derive the discrete state-space model. We define the state vector as \(\mathbf{x}_k = [SOC_k, U_{1,k}, U_{2,k}]^T\). The input is the measured current \(I_k\), and the output is the terminal voltage \(U_{t,k}\). By discretizing the continuous-time equations with a sampling period \(T\), we obtain:
$$
\begin{aligned}
\mathbf{x}_{k+1} &= \mathbf{A}_k \mathbf{x}_k + \mathbf{B}_k I_k + \mathbf{w}_k \\
U_{t,k} &= U_{oc}(SOC_k) – U_{1,k} – U_{2,k} – R_0 I_k + v_k \\
&= \mathbf{C}_k \mathbf{x}_k – R_0 I_k + v_k
\end{aligned}
$$
where \(\mathbf{w}_k\) and \(v_k\) represent the process and measurement noise, respectively. The matrices are defined as:
$$
\mathbf{A}_k = \begin{bmatrix}
1 & 0 & 0 \\
0 & e^{-T/(R_1 C_1)} & 0 \\
0 & 0 & e^{-T/(R_2 C_2)}
\end{bmatrix}, \quad
\mathbf{B}_k = \begin{bmatrix}
-\frac{\eta T}{C_n} \\
R_1(1 – e^{-T/(R_1 C_1)}) \\
R_2(1 – e^{-T/(R_2 C_2)})
\end{bmatrix}
$$
$$
\mathbf{C}_k = \left[ \frac{\partial U_{oc}(SOC_k)}{\partial SOC_k}, \ -1, \ -1 \right]
$$
The term \(\frac{\partial U_{oc}(SOC_k)}{\partial SOC_k}\) is the slope of the OCV-SOC curve, which is crucial for the filter’s correct operation. Therefore, an accurate characterization of the \(U_{oc}(SOC)\) relationship is essential. This is typically obtained through a low-current calibration test, where the battery is slowly discharged/charged with periodic rest periods to measure the equilibrium voltage. The relationship is highly nonlinear, especially at high and low SOC regions for lithium-ion batteries. A high-order polynomial is often used for fitting:
$$
U_{oc}(SOC) = a_9 SOC^9 + a_8 SOC^8 + … + a_1 SOC + a_0
$$
The model parameters \(R_0, R_1, C_1, R_2, C_2\) are not constant but vary with SOC, temperature, and aging. For a specific set of conditions, they can be identified using parameter identification techniques such as hybrid pulse power characterization (HPPC) tests. The table below summarizes a typical set of parameters identified for a lithium iron phosphate (LFP) cell at 25°C and a specific SOC point.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Ohmic Resistance | \(R_0\) | 0.0025 | Ω |
| Polarization Resistance 1 | \(R_1\) | 0.0012 | Ω |
| Polarization Capacitance 1 | \(C_1\) | 8500 | F |
| Polarization Resistance 2 | \(R_2\) | 0.0018 | Ω |
| Polarization Capacitance 2 | \(C_2\) | 180000 | F |
| Time Constant 1 | \(\tau_1 = R_1 C_1\) | ~10.2 | s |
| Time Constant 2 | \(\tau_2 = R_2 C_2\) | ~324 | s |
The H-Infinity Filter (HF) for SOC Estimation
The H-infinity filter is designed from a game theory or worst-case optimization perspective. Unlike the Kalman filter which minimizes the mean-square estimation error, the HF aims to minimize the worst-case amplification of the estimation error caused by disturbances (process noise, measurement noise, and initial condition errors), without requiring knowledge of their exact statistical properties.
Consider the discrete-time system defined earlier:
$$
\begin{aligned}
\mathbf{x}_{k+1} &= \mathbf{A}_k \mathbf{x}_k + \mathbf{B}_k u_k + \mathbf{w}_k \\
y_k &= \mathbf{C}_k \mathbf{x}_k + D u_k + v_k
$$
We want to estimate a linear combination of the state, \(\mathbf{z}_k = \mathbf{L}_k \mathbf{x}_k\) (for SOC estimation, \(\mathbf{L}_k = [1, 0, 0]\)). Let \(\hat{\mathbf{z}}_{k|k}\) be the estimate of \(\mathbf{z}_k\) given measurements up to time \(k\). The performance criterion for the H-infinity filter is defined as:
$$
J = \frac{\sum_{k=0}^{N-1} \|\mathbf{z}_k – \hat{\mathbf{z}}_{k|k}\|_{\mathbf{S}_k}^2}{\|\mathbf{x}_0 – \hat{\mathbf{x}}_0\|_{\mathbf{P}_0^{-1}}^2 + \sum_{k=0}^{N-1} \left( \|\mathbf{w}_k\|_{\mathbf{Q}_k^{-1}}^2 + \|v_k\|_{R_k^{-1}}^2 \right)}
$$
where \(\mathbf{S}_k\) is a user-defined positive definite weighting matrix, \(\mathbf{P}_0\), \(\mathbf{Q}_k\), and \(R_k\) are positive definite matrices reflecting confidence in the initial guess and the disturbances. The notation \(\|\mathbf{a}\|_{\mathbf{M}}^2\) denotes \(\mathbf{a}^T \mathbf{M} \mathbf{a}\).
The objective of the H-infinity filter is to find an estimation strategy such that the supremum (worst-case) of \(J\) over all admissible disturbances is less than a prescribed bound \(\gamma^2\) (where \(\gamma = 1/\sqrt{\theta}\) and \(\theta\) is the performance level):
$$
\sup_{\mathbf{x}_0, \mathbf{w}, \mathbf{v}} J < \gamma^2
$$
This is equivalent to ensuring:
$$
\sum_{k=0}^{N-1} \|\mathbf{z}_k – \hat{\mathbf{z}}_{k|k}\|_{\mathbf{S}_k}^2 < \gamma^2 \left[ \|\mathbf{x}_0 – \hat{\mathbf{x}}_0\|_{\mathbf{P}_0^{-1}}^2 + \sum_{k=0}^{N-1} \left( \|\mathbf{w}_k\|_{\mathbf{Q}_k^{-1}}^2 + \|v_k\|_{R_k^{-1}}^2 \right) \right]
$$
for all nonzero sequences \(\{\mathbf{w}_k\}\), \(\{v_k\}\), and initial state error.
Solving this minimax problem leads to the following recursive H-infinity filtering algorithm, which resembles the Kalman filter but with a modified gain and covariance update:
HF Algorithm Steps:
1. Initialization: Set \(\hat{\mathbf{x}}_{0|0}\), \(\mathbf{P}_{0|0} > 0\). Choose \(\theta\) (performance bound parameter).
2. State Prediction:
$$
\hat{\mathbf{x}}_{k|k-1} = \mathbf{A}_{k-1} \hat{\mathbf{x}}_{k-1|k-1} + \mathbf{B}_{k-1} u_{k-1}
$$
3. Innovation:
$$
e_k = y_k – (\mathbf{C}_k \hat{\mathbf{x}}_{k|k-1} + D u_k)
$$
4. Gain Calculation: A critical step where the robustness is embedded.
$$
\mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{C}_k^T \left[ \mathbf{C}_k \mathbf{P}_{k|k-1} \mathbf{C}_k^T + R_k – \theta \mathbf{C}_k \mathbf{P}_{k|k-1} \mathbf{S}_k \mathbf{P}_{k|k-1} \mathbf{C}_k^T \right]^{-1}
$$
Note the extra term involving \(\theta\) and \(\mathbf{S}_k\) compared to the Kalman gain.
5. State Update:
$$
\hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k e_k
$$
6. Covariance Update:
$$
\mathbf{P}_{k|k} = \mathbf{P}_{k|k-1} – \mathbf{P}_{k|k-1} \left[ \mathbf{C}_k^T \ \ \mathbf{H}_k^T \right] \mathbf{R}_e^{-1} \begin{bmatrix} \mathbf{C}_k \\ \mathbf{H}_k \end{bmatrix} \mathbf{P}_{k|k-1}
$$
where \(\mathbf{H}_k = \sqrt{\theta} \mathbf{S}_k^{1/2}\) and \(\mathbf{R}_e = \begin{bmatrix} R_k & 0 \\ 0 & -\mathbf{I} \end{bmatrix} + \begin{bmatrix} \mathbf{C}_k \\ \mathbf{H}_k \end{bmatrix} \mathbf{P}_{k|k-1} \left[ \mathbf{C}_k^T \ \mathbf{H}_k^T \right]\). A more common and computationally stable form is the Riccati equation:
$$
\mathbf{P}_{k|k}^{-1} = \mathbf{P}_{k|k-1}^{-1} + \mathbf{C}_k^T R_k^{-1} \mathbf{C}_k – \theta \mathbf{S}_k
$$
with the prediction covariance given by:
$$
\mathbf{P}_{k+1|k} = \mathbf{A}_k \mathbf{P}_{k|k} \mathbf{A}_k^T + \mathbf{Q}_k
$$
7. SOC Extraction: The estimated SOC is the first element of \(\hat{\mathbf{x}}_{k|k}\).
$$
\widehat{SOC}_k = [1, 0, 0] \cdot \hat{\mathbf{x}}_{k|k}
$$
The filter requires tuning parameters \(\theta\), \(\mathbf{S}_k\), \(\mathbf{Q}_k\), and \(R_k\). The parameter \(\theta\) must be chosen such that \(\mathbf{P}_{k|k-1}^{-1} – \theta \mathbf{S}_k + \mathbf{C}_k^T R_k^{-1} \mathbf{C}_k > 0\) to ensure a positive definite solution. A smaller \(\theta\) provides better robustness (tighter bound on \(J\)) but may lead to numerical issues if too small. For SOC estimation of a lithium-ion battery, \(\mathbf{S}_k\) is often set to penalize the SOC estimation error specifically, e.g., \(\mathbf{S}_k = \text{diag}([1, 0, 0])\).
The Proposed Strong Tracking H-Infinity Filter (STHF)
While the HF provides excellent robustness against noise uncertainty, its structure is inherently “conservative.” The covariance update, designed for a worst-case disturbance profile, may not react swiftly enough when the actual system undergoes an abrupt change not modeled by the nominal process noise covariance \(\mathbf{Q}_k\). This can happen with a lithium-ion battery during a sudden high-current pulse, a rapid temperature change, or when model parameters deviate significantly from their identified values.
The Strong Tracking Filter (STF) concept, originally developed for the Kalman filter, addresses this by introducing a time-varying fading factor \(\lambda_k\) (with \(\lambda_k \geq 1\)) into the prediction covariance equation. This factor artificially inflates the predicted covariance, which in turn increases the filter gain, making the filter more responsive to the latest measurement and enabling it to “catch up” with a rapidly changing state.
We innovatively integrate this concept into the HF framework to create the Strong Tracking H-infinity Filter (STHF). The key modification is applied to the covariance prediction step. The standard HF prediction is \(\mathbf{P}_{k+1|k} = \mathbf{A}_k \mathbf{P}_{k|k} \mathbf{A}_k^T + \mathbf{Q}_k\). The STHF modifies this to:
$$
\mathbf{P}_{k+1|k} = \lambda_{k+1} \mathbf{A}_k \mathbf{P}_{k|k} \mathbf{A}_k^T + \mathbf{Q}_k
$$
The challenge lies in calculating \(\lambda_{k+1}\) optimally. The principle is to force the innovation sequence \(e_k\) to remain orthogonal or “white,” which is a condition for optimal filtering. When the system model is perfect and the filter is optimal, the innovation sequence is a white noise process. If a state change occurs, the innovations become correlated. The fading factor is adjusted to restore this orthogonality.
The calculation of \(\lambda_{k+1}\) proceeds as follows:
1. Compute the actual innovation covariance:
$$
\mathbf{V}_k = E[e_k e_k^T] \approx \frac{1}{k} \sum_{i=1}^{k} e_i e_i^T
$$
In practice, a recursive forgetting factor \(\rho\) (e.g., 0.95) is used for online computation:
$$
\mathbf{V}_k = \frac{\rho \mathbf{V}_{k-1} + e_k e_k^T}{1 + \rho}
$$
with \(\mathbf{V}_0 = e_0 e_0^T\).
2. The theoretical innovation covariance based on the current filter parameters is:
$$
\mathbf{\Sigma}_k = \mathbf{C}_k \mathbf{P}_{k|k-1} \mathbf{C}_k^T + R_k
$$
3. For the STF principle to hold, we want \(\mathbf{V}_k \approx \mathbf{\Sigma}_k\). By substituting the modified prediction equation, we get a relation involving \(\lambda_k\). Solving for a scalar factor (applied to the part of the covariance originating from the state transition) leads to:
$$
\lambda_k = \max\left(1, \frac{\text{tr}(\mathbf{N}_k)}{\text{tr}(\mathbf{M}_k)}\right)
$$
where:
$$
\begin{aligned}
\mathbf{N}_k &= \mathbf{V}_k – \mathbf{C}_k \mathbf{Q}_{k-1} \mathbf{C}_k^T – R_k \\
\mathbf{M}_k &= \mathbf{C}_k \mathbf{A}_{k-1} \mathbf{P}_{k-1|k-1} \mathbf{A}_{k-1}^T \mathbf{C}_k^T
\end{aligned}
$$
Here, \(\text{tr}(\cdot)\) denotes the matrix trace. The term \(\mathbf{N}_k\) represents the residual covariance not explained by the nominal process/measurement noise. If \(\mathbf{N}_k\) is larger than \(\mathbf{M}_k\), \(\lambda_k > 1\), inflating the predicted covariance and boosting the filter gain.
The complete STHF algorithm for lithium-ion battery SOC estimation merges the HF robustness with this adaptive fading mechanism. The step-by-step procedure is summarized in the table below, highlighting the key differences from the standard HF.
| Step | H-Infinity Filter (HF) | Strong Tracking H-Infinity Filter (STHF) | Purpose |
|---|---|---|---|
| 1. Initialization | Set \(\hat{\mathbf{x}}_{0}\), \(\mathbf{P}_{0}\), \(\theta\), \(\mathbf{S}\), \(\mathbf{Q}\), \(R\) | Set \(\hat{\mathbf{x}}_{0}\), \(\mathbf{P}_{0}\), \(\theta\), \(\mathbf{S}\), \(\mathbf{Q}\), \(R\), \(\rho\) | Define initial state and tuning parameters. |
| 2. State Prediction | \(\hat{\mathbf{x}}_{k|k-1} = \mathbf{A} \hat{\mathbf{x}}_{k-1} + \mathbf{B} I_{k-1}\) | \(\hat{\mathbf{x}}_{k|k-1} = \mathbf{A} \hat{\mathbf{x}}_{k-1} + \mathbf{B} I_{k-1}\) | Predict state based on model. |
| 3. Covariance Prediction | \(\mathbf{P}_{k|k-1} = \mathbf{A} \mathbf{P}_{k-1} \mathbf{A}^T + \mathbf{Q}\) | Calculate \(\lambda_k\) using Eq. (X). \(\mathbf{P}_{k|k-1} = \lambda_k \mathbf{A} \mathbf{P}_{k-1} \mathbf{A}^T + \mathbf{Q}\) |
Predict error covariance. STHF inflates it via \(\lambda_k \geq 1\). |
| 4. Innovation | \(e_k = U_{t,k} – (U_{oc}(\widehat{SOC}_{k|k-1}) – \hat{U}_{1,k|k-1} – \hat{U}_{2,k|k-1} – R_0 I_k)\) | Same as HF. | Difference between measured and predicted voltage. |
| 5. Update Innovation Covariance | Not typically used. | \(\mathbf{V}_k = (\rho \mathbf{V}_{k-1} + e_k e_k^T)/(1+\rho)\) | Online estimate of actual innovation sequence power (for STHF only). |
| 6. Gain Calculation | \(\mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{C}_k^T [\mathbf{C}_k \mathbf{P}_{k|k-1} \mathbf{C}_k^T + R – \theta \mathbf{C}_k \mathbf{P}_{k|k-1} \mathbf{S} \mathbf{P}_{k|k-1} \mathbf{C}_k^T]^{-1}\) | Same formula, but uses the inflated \(\mathbf{P}_{k|k-1}\) from Step 3. | Compute optimal filter gain. STHF gain is larger when \(\lambda_k > 1\). |
| 7. State Update | \(\hat{\mathbf{x}}_{k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k e_k\) | Same as HF. | Correct state prediction with measurement. |
| 8. Covariance Update | \(\mathbf{P}_{k} = \mathbf{P}_{k|k-1} – \mathbf{P}_{k|k-1} [\mathbf{C}_k^T \ \mathbf{H}^T] \mathbf{R}_e^{-1} [\mathbf{C}_k; \mathbf{H}] \mathbf{P}_{k|k-1}\) | Same as HF. | Update error covariance after measurement. |
| 9. SOC Output | \(\widehat{SOC}_k = [1, 0, 0] \cdot \hat{\mathbf{x}}_{k}\) | Same as HF. | Extract the estimated SOC from the state vector. |
This STHF algorithm maintains all the robustness properties of the HF against noise uncertainty. Simultaneously, the adaptive fading factor \(\lambda_k\) provides a self-tuning mechanism. Under normal, steady conditions, \(\lambda_k\) will be close to 1, and the STHF behaves almost identically to the standard HF. When an abrupt change occurs in the lithium-ion battery’s dynamics—causing the innovation sequence to become persistently large—\(\lambda_k\) increases significantly. This inflates the predicted covariance \(\mathbf{P}_{k|k-1}\), leading to a larger gain matrix \(\mathbf{K}_k\). The larger gain assigns more weight to the incoming measurement, enabling the filter to rapidly adjust its state estimate to track the true system state. Once the change has been tracked, the innovation sequence becomes white again, and \(\lambda_k\) returns to near 1.
Experimental Validation and Results Analysis
To validate the performance of the proposed STHF algorithm for lithium-ion battery SOC estimation, comprehensive experimental tests were conducted. A commercial 36 Ah Lithium Iron Phosphate (LFP) prismatic cell was used. The test bench comprised a high-precision battery cycler (BTS-4000) for applying current profiles and logging voltage/current data, a thermal chamber to maintain constant temperature (25°C), and a host computer running MATLAB/Simulink for algorithm implementation and data analysis.
The first step was model parameterization. A Hybrid Pulse Power Characterization (HPPC) test was performed at various SOC levels (from 100% to 0% in 10% steps) to identify the parameters \(R_0(SOC)\), \(R_1(SOC)\), \(C_1(SOC)\), \(R_2(SOC)\), \(C_2(SOC)\). The OCV-SOC relationship was obtained through a low-current (C/20) calibration test. A 9th-order polynomial was fitted to the OCV data, as shown in the equation in Section 2, providing a highly accurate \(U_{oc}(SOC)\) and its derivative \(\partial U_{oc}/\partial SOC\).
Two distinct dynamic load profiles were used to evaluate the algorithms:
1. Dynamic Stress Test (DST) Profile: This is a standard simplified driving cycle from the USABC manual. It consists of a series of constant-current discharge and charge pulses with varying magnitudes and durations, followed by rest periods. It is highly dynamic and stresses the model’s ability to handle rapid current changes, making it an excellent benchmark for SOC estimation algorithms.
2. Intermittent Discharge Profile: This profile consists of constant-current discharge pulses (e.g., 1C rate) each followed by a substantial rest period. It tests the algorithm’s ability to accurately integrate current during discharge and then correctly estimate the voltage recovery during rest, which is critical for validating the polarization dynamics of the model.
For both tests, the initial SOC was set to 100%. The “true” SOC reference was calculated using high-accuracy ampere-hour integration with a calibrated current sensor, which serves as the ground truth for performance comparison. Both the standard HF and the proposed STHF were implemented with the same initial conditions and tuning parameters (\(\mathbf{P}_0\), \(\mathbf{Q}\), \(R\), \(\theta\), \(\mathbf{S}\)) to ensure a fair comparison. The forgetting factor \(\rho\) for the STHF was set to 0.95.
The performance metrics evaluated include:
- SOC Estimation Error: \(Error(\%) = \widehat{SOC} – SOC_{true}\)
- Root Mean Square Error (RMSE): \(\text{RMSE} = \sqrt{\frac{1}{N}\sum_{k=1}^{N} (\widehat{SOC}_k – SOC_{true,k})^2}\)
- Maximum Absolute Error (MAE): \(\text{MAE} = \max(|\widehat{SOC}_k – SOC_{true,k}|)\)
| Test Profile | Algorithm | RMSE (%) | MAE (%) | Error Range (%) |
|---|---|---|---|---|
| DST Cycle | Standard HF | 1.15 | 3.50 | -3.50 to +0.20 |
| Proposed STHF | 0.85 | 2.10 | -2.10 to -1.80 | |
| Intermittent Discharge | Standard HF | 0.62 | 1.40 | -0.20 to +1.40 |
| Proposed STHF | 0.08 | 0.20 | -0.20 to +0.20 |
Analysis of DST Cycle Results: The DST profile, with its aggressive charge-discharge pulses, presents a significant challenge. The results clearly demonstrate the advantage of the STHF. The standard HF showed a noticeable tracking lag, especially after strong regenerative braking (charge) pulses, leading to a maximum error of -3.5%. The error range was asymmetric (-3.5% to +0.2%), indicating a consistent underestimation bias during highly dynamic phases. In contrast, the proposed STHF algorithm significantly reduced this lag. Its error range was tighter and more symmetric (-2.1% to -1.8%), and both the RMSE (0.85% vs 1.15%) and MAE (2.1% vs 3.5%) were substantially lower. This improvement is directly attributable to the fading factor \(\lambda_k\), which increased during sudden current transitions, boosting the filter gain and allowing faster correction of the state estimate.
Analysis of Intermittent Discharge Results: This profile tests the model’s steady-state accuracy and the filter’s behavior during rest periods. The standard HF performed reasonably well but exhibited a noticeable drift as the test progressed, culminating in a +1.4% error at the end of discharge. This is likely due to small but cumulative model inaccuracies or slight current sensor bias that the conservative HF could not fully correct. The STHF, however, delivered exceptional performance. Its error remained tightly bounded within ±0.2% throughout the entire discharge process, resulting in a remarkably low RMSE of 0.08%. This indicates that the STHF not only handles transients better but also maintains superior steady-state accuracy. The adaptive mechanism appears to help fine-tune the filter’s responsiveness even to slow, accumulating discrepancies, preventing long-term drift.
The terminal voltage prediction is another important indicator. Both algorithms predicted voltage within a small error band (±1.5% for DST, ±1.5% for intermittent), confirming the underlying model’s validity. However, the STHF consistently showed slightly lower voltage prediction errors, particularly during the transient phases of the DST cycle, corroborating its improved state tracking capability.
Conclusion and Future Perspectives
Accurate and robust State-of-Charge estimation remains a critical challenge for the advancement of lithium-ion battery management technology. In this work, we have addressed a specific limitation of the robust H-infinity filter—its sluggish response to abrupt state changes—by developing a novel Strong Tracking H-infinity Filter (STHF). The proposed algorithm ingeniously incorporates the fading factor mechanism from Strong Tracking Filter theory into the H-infinity framework. This factor is computed online based on the innovation sequence, allowing the filter to autonomously increase its gain when the actual system behavior deviates from the model, thereby achieving rapid tracking of dynamic transients while preserving the HF’s inherent robustness against noise and model uncertainty.
Through rigorous experimental validation using a second-order RC model for a lithium-ion battery, the STHF algorithm demonstrated superior performance compared to the standard HF. Under the highly dynamic DST profile, the STHF reduced the maximum SOC error from 3.5% to 2.1% and tightened the error range. More impressively, under an intermittent discharge profile, it virtually eliminated estimation drift, confining the error to within ±0.2%, a significant improvement over the standard HF’s +1.4% error. These results conclusively prove that the STHF algorithm enhances both the dynamic tracking capability and the steady-state accuracy of SOC estimation for lithium-ion batteries.
The implications of this work are significant for practical BMS design. The STHF offers a more reliable and accurate SOC estimate under real-world driving conditions, which are characterized by unpredictable and aggressive load variations. This leads to better vehicle range prediction, more efficient energy management, and enhanced battery protection.
Future research can extend this work in several promising directions. First, the algorithm should be tested and validated across a wider range of temperatures and aging states of the lithium-ion battery, with adaptive mechanisms for updating model parameters. Second, the computational efficiency of the STHF should be analyzed for implementation on embedded BMS hardware. Finally, the core idea could be extended to joint estimation of other critical states, such as State-of-Health (SOH) and State-of-Power (SOP), creating a unified, robust multi-state estimator for next-generation intelligent BMS. The STHF algorithm represents a meaningful step forward in the pursuit of ultra-reliable and precise battery state estimation.